Welcome to your first assignment of this week!
Because word embeddings are very computionally expensive to train, most ML practitioners will load a pre-trained set of embeddings.
After this assignment you will be able to:
Let's get started! Run the following cell to load the packages you will need.
import numpy as np
from w2v_utils import *
Using TensorFlow backend.
Next, lets load the word vectors. For this assignment, we will use 50-dimensional GloVe vectors to represent words. Run the following cell to load the word_to_vec_map.
words, word_to_vec_map = read_glove_vecs('data/glove.6B.50d.txt')
You've loaded:
words: set of words in the vocabulary.word_to_vec_map: dictionary mapping words to their GloVe vector representation.You've seen that one-hot vectors do not do a good job cpaturing what words are similar. GloVe vectors provide much more useful information about the meaning of individual words. Lets now see how you can use GloVe vectors to decide how similar two words are.
To measure how similar two words are, we need a way to measure the degree of similarity between two embedding vectors for the two words. Given two vectors $u$ and $v$, cosine similarity is defined as follows:
$$\text{CosineSimilarity(u, v)} = \frac {u . v} {||u||_2 ||v||_2} = cos(\theta) \tag{1}$$where $u.v$ is the dot product (or inner product) of two vectors, $||u||_2$ is the norm (or length) of the vector $u$, and $\theta$ is the angle between $u$ and $v$. This similarity depends on the angle between $u$ and $v$. If $u$ and $v$ are very similar, their cosine similarity will be close to 1; if they are dissimilar, the cosine similarity will take a smaller value.

Exercise: Implement the function cosine_similarity() to evaluate similarity between word vectors.
Reminder: The norm of $u$ is defined as $ ||u||_2 = \sqrt{\sum_{i=1}^{n} u_i^2}$
# GRADED FUNCTION: cosine_similarity
def cosine_similarity(u, v):
"""
Cosine similarity reflects the degree of similariy between u and v
Arguments:
u -- a word vector of shape (n,)
v -- a word vector of shape (n,)
Returns:
cosine_similarity -- the cosine similarity between u and v defined by the formula above.
"""
distance = 0.0
### START CODE HERE ###
# Compute the dot product between u and v (≈1 line)
dot = np.sum(u *v)
print(dot)
# Compute the L2 norm of u (≈1 line)
norm_u = np.linalg.norm(u,2)
# Compute the L2 norm of v (≈1 line)
norm_v = np.linalg.norm(v,2)
# Compute the cosine similarity defined by formula (1) (≈1 line)
cosine_similarity = dot / (norm_u * norm_v)
### END CODE HERE ###
return cosine_similarity
father = word_to_vec_map["father"]
mother = word_to_vec_map["mother"]
ball = word_to_vec_map["ball"]
crocodile = word_to_vec_map["crocodile"]
france = word_to_vec_map["france"]
italy = word_to_vec_map["italy"]
paris = word_to_vec_map["paris"]
rome = word_to_vec_map["rome"]
print("cosine_similarity(father, mother) = ", cosine_similarity(father, mother))
print("cosine_similarity(ball, crocodile) = ",cosine_similarity(ball, crocodile))
print("cosine_similarity(france - paris, rome - italy) = ",cosine_similarity(france - paris, rome - italy))
26.6744302608 cosine_similarity(father, mother) = 0.890903844289 7.33508827507 cosine_similarity(ball, crocodile) = 0.274392462614 -9.35047910933 cosine_similarity(france - paris, rome - italy) = -0.675147930817
Expected Output:
| **cosine_similarity(father, mother)** = | 0.890903844289 |
| **cosine_similarity(ball, crocodile)** = | 0.274392462614 |
| **cosine_similarity(france - paris, rome - italy)** = | -0.675147930817 |
After you get the correct expected output, please feel free to modify the inputs and measure the cosine similarity between other pairs of words! Playing around the cosine similarity of other inputs will give you a better sense of how word vectors behave.
In the word analogy task, we complete the sentence "*a* is to *b* as *c* is to **____**". An example is '*man* is to *woman* as *king* is to *queen*' . In detail, we are trying to find a word d, such that the associated word vectors $e_a, e_b, e_c, e_d$ are related in the following manner: $e_b - e_a \approx e_d - e_c$. We will measure the similarity between $e_b - e_a$ and $e_d - e_c$ using cosine similarity.
Exercise: Complete the code below to be able to perform word analogies!
# GRADED FUNCTION: complete_analogy
def complete_analogy(word_a, word_b, word_c, word_to_vec_map):
"""
Performs the word analogy task as explained above: a is to b as c is to ____.
Arguments:
word_a -- a word, string
word_b -- a word, string
word_c -- a word, string
word_to_vec_map -- dictionary that maps words to their corresponding vectors.
Returns:
best_word -- the word such that v_b - v_a is close to v_best_word - v_c, as measured by cosine similarity
"""
# convert words to lower case
word_a, word_b, word_c = word_a.lower(), word_b.lower(), word_c.lower()
### START CODE HERE ###
# Get the word embeddings v_a, v_b and v_c (≈1-3 lines)
e_a, e_b, e_c = word_to_vec_map[word_a], word_to_vec_map[word_b], word_to_vec_map[word_c]
### END CODE HERE ###
words = word_to_vec_map.keys()
max_cosine_sim = -100 # Initialize max_cosine_sim to a large negative number
best_word = None # Initialize best_word with None, it will help keep track of the word to output
# loop over the whole word vector set
for w in words:
# to avoid best_word being one of the input words, pass on them.
if w in [word_a, word_b, word_c] :
continue
### START CODE HERE ###
# Compute cosine similarity between the vector (e_b - e_a) and the vector ((w's vector representation) - e_c) (≈1 line)
cosine_sim = cosine_similarity((e_b - e_a),(word_to_vec_map[w] - e_c))
# If the cosine_sim is more than the max_cosine_sim seen so far,
# then: set the new max_cosine_sim to the current cosine_sim and the best_word to the current word (≈3 lines)
if cosine_sim > max_cosine_sim:
max_cosine_sim = cosine_sim
best_word = w
### END CODE HERE ###
return best_word
Run the cell below to test your code, this may take 1-2 minutes.
triads_to_try = [('italy', 'italian', 'spain'), ('india', 'delhi', 'japan'), ('man', 'woman', 'boy'), ('small', 'smaller', 'large')]
for triad in triads_to_try:
print ('{} -> {} :: {} -> {}'.format( *triad, complete_analogy(*triad,word_to_vec_map)))
3.94549512422 3.80373866687 3.37722599357 3.46150798285 1.69586007213 4.08280238925 2.44141146709 7.70628448759 7.34183695946 3.73616568836 4.25650217807 5.22977655439 3.5825891318 3.22615241404 4.03221530596 5.46938836726 3.46326404987 5.08797439034 5.2481201849 5.75159003345 2.08627738373 4.81891353682 2.63810825034 3.72677338747 3.52799220633 3.81536227484 6.1031940473 4.78420819513 5.48577106489 7.89825659902 2.94487759041 3.10551584584 3.23966008637 2.75337073203 3.05861715703 4.58388005526 2.88539305672 3.01701528415 6.39978285566 2.95104762458 3.62187074855 2.92571354376 3.11407856023 -0.535572793116 3.44263613635 3.74672182477 5.70307169046 3.00679544482 5.25134105722 2.42058672544 4.06541022557 4.220266687 4.44127012018 1.1642329696 1.36263176023 4.35182519369 4.26689349274 4.67690383204 4.35993954121 4.04983206595 2.60429704842 2.65128080881 1.33113368787 3.46998595106 2.67601592828 2.37708566431 2.46924956198 4.92807951674 5.12961248874 3.81582413258 2.25710365216 5.6447515622 1.74571731583 3.80110578338 1.09931441266 3.18540297846 1.73431920664 5.14723334289 4.41960152836 2.50415296853 3.16482552055 3.83391107941 3.15821371594 1.39176999601 4.55234757415 0.808705686954 2.30776469806 4.01549678814 2.77997776604 3.62935776434 3.44407973502 2.79722663754 1.80096679041 0.268654350506 1.1006675755 1.87093403129 5.1601651175 3.33088181327 -0.128932012944 2.06875788317 2.99359536006 2.91224409106 2.90730381221 4.34482712771 -0.643438824134 3.93704329279 1.40972668809 1.50833851243 1.75658120021 5.46485102744 2.96800230511 2.73250381083 -0.583450076722 2.31598945669 1.89979530382 2.26513359712 4.83388557197 5.465303468 0.916687873986 2.7059455667 2.17251225093 4.33148470358 2.77947118794 2.3525814803 3.56116632525 5.41764843771 1.49524841249 2.43974905327 6.39784329396 6.6538210826 4.03155496957 1.75604460283 -0.203942649225 4.04581950693 5.03011042549 2.22994521071 3.52806141537 1.31292243188 1.73322520747 -0.113028072476 7.94306208653 6.31679030548 7.04961844214 4.74684651314 5.52739842931 3.15448109486 3.12683697117 2.23766111471 3.73516813917 2.65213837492 3.04096290464 3.35042679716 3.64605269454 3.04572609925 6.38079814993 2.9817732478 0.854120532793 7.16295314204 4.05939215617 2.70679202178 -0.36863055011 5.9939834945 2.391468188 2.75770574655 5.6833103377 5.18055822772 5.37100612187 7.61904443555 4.53862833216 2.90412428385 5.69835049568 2.66460402644 5.96462319982 2.9987346068 3.4313680061 5.80438031829 1.03912937957 2.62100036591 6.39912290311 2.04304855016 7.03854994299 2.81555056038 0.174025051789 2.8449249646 2.83202930364 2.48175436647 1.10674958218 2.54686801075 3.9481319 0.941044001546 1.75498024846 4.75541304329 5.22066347527 0.174966117289 2.87464228015 -0.113464094728 4.45242964165 3.34947795718 4.64298068581 4.67349947635 5.13535292523 6.23661310427 2.77744178711 3.83997382233 2.18246628268 6.84367780969 2.00208799765 1.80442907199 2.02839232709 5.40361631364 4.21681351803 3.21521506082 3.16941519122 6.18798794418 5.08656763878 4.51305280863 5.19815870266 1.83837646786 2.97385861922 4.76230992905 5.57374979868 3.7190832547 0.659739818206 4.20792046259 3.59696811739 7.73630874351 -1.4157159205 1.01022907421 3.54223933131 2.39858795469 1.69284090811 3.31744097509 1.02358487404 -0.273300108594 5.8841712664 5.99595640667 3.60783775343 2.29308903366 5.36311579611 6.45775986203 1.86267851716 1.1225658453 1.54137568932 6.59954836944 4.36382809413 1.18187976376 3.67087413175 3.38233732767 2.37893550413 3.2568706681 7.23236851225 0.811122492877 -1.33983423303 2.03734471233 5.51842637733 3.82496029252 6.24237076622 2.78404497538 5.90965363931 -0.147085471166 2.32046573902 5.38735335478 4.60893224901 3.29773420002 4.65797668911 1.95487234836 2.82739349972 3.52716840543 2.24136237933 2.67162037941 5.77336960989 4.42664556328 1.37776245419 4.66808056481 4.03025577582 1.09904163008 6.40672836092 1.30632373869 4.39331943383 5.44152296001 3.29708777569 3.96640474203 1.46931052005 4.62362909499 2.96071197654 4.13706683025 2.02284866153 -0.27929293379 5.0901992897 1.10069494753 0.109958867754 2.80758082543 2.47538898044 5.69010881153 -0.311803432938 2.93303550726 2.92394025281 8.31623661034 7.15866107989 5.25031148097 8.23251582768 4.39995831965 5.82959355337 2.50564387938 1.82731336289 3.51905525414 4.56306204279 2.60411296043 3.45226689985 4.39503295023 0.25720103136 6.77306969598 1.90947835309 2.00907980065 0.525598539938 1.12582084837 4.52650464919 4.13771498532 4.85701193635 6.61906933872 -1.84307870691 3.38785450149 -0.615400963358 1.40435763513 2.6772015584 4.60782823093 2.13534866208 7.19199894583 3.84798763846 5.84861596927 7.20531360295 2.56423394555 2.49949397277 5.06163148501 3.78945230688 4.97432235582 4.1228794439 4.85464161108 3.55601012332 2.66634201967 4.07286717588 1.4116962949 7.21905324077 1.76021868404 1.20636676777 4.57057362631 3.81074097947 3.77842577485 9.3538396837 1.11821050305 3.37269439653 2.11614066757 4.02140589211 4.30997997148 4.70376794825 2.12506102927 7.07698039114 6.66310878537 4.63341819868 4.17954347641 3.36958733607 0.87702416795 5.7951835474 2.58743573103 3.2992747959 3.65860017123 -0.513195596689 0.712301960378 -0.147432615946 6.18568779498 1.74332989452 5.11739643661 1.32405761576 3.099463637 4.26359790541 1.61297645822 3.74446839736 0.999686231086 0.565690773936 5.52967915124 2.24288179078 3.14849784803 1.37985760614 2.4783293132 1.67846739139 2.52002968553 0.948921355839 0.954859663184 3.20210422164 2.02312060539 1.33284425244 5.06820029994 4.75389640805 6.58444996837 2.06235272231 3.07712046823 1.91248309374 3.79026579993 2.32143085938 4.18545061686 3.25367710475 2.39548755293 1.3205489681 9.1490189201 1.89430570365 3.40297301062 7.66468189893 3.01261575341 1.77238531415 0.789806040148 0.867897387454 1.74125322407 -0.882045000056 3.41946939893 2.1787592549 3.39006807165 3.08004256853 2.97153252568 1.53009318866 5.52214693949 0.697856216796 2.21785169389 0.838302904964 4.54523409392 1.73513442917 -0.274917411 6.06875557065 1.68000832405 4.56187305782 5.4285141613 3.62466625546 4.40471827695 4.91959243287 3.77478646597 2.03957355677 2.05756742743 3.73330677423 2.51216744422 3.17342201395 2.68005217496 3.93858927141 1.9572854148 3.25184676453 1.67659239753 4.36996584644 1.80176426843 3.28777607852 4.79553672923 0.668390261312 5.99551586744 2.39182663531 0.959912314086 4.16692955389 2.43551352476 4.64667828772 2.56271722661 6.4596706627 3.04003668108 7.6571866156 6.36362446749 3.04688943542 7.97602781431 3.23312989057 5.76681168869 5.15909406015 0.787410742828 2.78152448179 8.40479899485 2.77926986763 -0.481920875286 2.09541359942 -1.46901208604 4.78100825215 2.54157225401 -0.018882290394 4.68435912071 4.5257397117 1.79285246286 2.23664856972 3.88015520138 3.67590655256 2.79713783101 2.5888143541 0.367252878442 5.95055418683 1.37729394002 3.83977256345 2.00390823417 5.714428371 0.00247097663 1.22531908041 2.91414464325 3.80354465285 6.78041967055 4.89664920705 2.83803932116 6.19543772703 3.72643790294 1.21978468461 4.72960137434 7.05775303848 4.19860481055 3.44471329149 5.80723638864 1.67513955785 2.647098692 4.85867269095 7.23057060764 2.39424536661 -1.43773097123 2.07991103295 1.84680891039 3.3771737473 1.44634116155 8.33600929167 5.43206056885 4.78484040319 1.5944571836 3.55919509461 1.6488223561 4.93546490034 7.75209495747 4.96073431332 6.53930436327 0.708417133464 0.186057191209 3.05200497757 3.80854621674 1.14802814693 1.31072191676 3.02936233311 1.25394648425 3.72219539083 2.99601248533 5.28269270608 4.24799073922 6.88094753272 7.11071695284 -0.091073549491 6.66898764592 2.30810578119 3.97188914955 4.28423599185 6.05664418145 3.33547845761 -0.625145564442 3.11582950846 1.48368381642 -0.517537076198 4.50785362553 -1.6018504076 3.81978314694 -0.776680858948 3.21737169588 1.25385259548 1.85041267627 2.31650582913 5.8502964851 3.88147966018 1.70067550227 3.56712722263 2.26687811477 3.98412564336 0.0923874838327 8.05958816888 4.83182070372 3.70053711814 4.41523863797 4.76788406366 6.8820758664 5.2448481319 2.47359415685 1.67317663086 2.83710690559 -0.345371676038 2.99117859962 4.17357517224 3.83645770798 1.86181894152 1.8972428194 5.06432988968 1.71323263184 2.9030528098 2.43536485343 1.69421391109 1.83977543306 -0.50538854438 3.7067821116 2.93851789995 4.56434967971 1.6434336998 3.66280450082 6.1233170319 2.57960143488 5.25683401227 2.62098945077 3.91833324436 7.52948381446 6.90918684304 0.997877588329 2.88329585666 1.0010324723 3.38822122802 -1.42263661876 0.937240929572 3.90542717705 3.54878843512 3.11389816216 3.69421162448 2.68725672286 3.30063894857 1.93751731646 4.46778521181 -0.718979132486 3.76103170345 2.76114526083 4.50160617657 2.64054216012 4.34949483866 1.00824902525 1.50246456517 1.57921941185 6.88485314937 8.98557037674 1.19265357859 1.12303124654 3.50678862559 2.85913375086 4.17962488154 7.44302221703 7.00429938246 4.60374802965 0.536300992096 4.63116008539 3.54177327307 5.71045355455 4.30427443901 -2.09203429098 2.87504658306 6.07073228811 7.85766888224 4.55753045959 3.73934742051 4.58172616734 -0.176687475631 3.62324885402 4.14015153077 0.676012107424 4.77786553773 3.92537141642 2.69091269711 -2.02895645791 0.92823603946 4.24676058013 2.90048000031 1.18093792538 -0.737296478202 1.60611257416 4.33957830516 5.30907214416 0.595345986347 3.44028629742 4.44903273942 3.64832782566 5.41085811811 1.87212584899 7.04756268143 1.31302563724 3.22300750615 4.41899631259 5.11221767995 1.46441051259 7.47453582586 1.15284924403 4.23928371872 4.58337812386 1.96036483485 0.798684894034 3.01769094561 1.35175434114 6.98873260492 8.4951614821 3.84589646099 4.57702182978 4.88365808779 4.4781839359 0.495987343006 5.72202247165 4.09621932957 0.834623724882 4.01282402697 2.97871088292 3.5102456406 4.17334554834 0.901066270822 0.502717561155 1.96174189959 4.20376061329 2.3887542472 2.88331451072 1.18862627151 1.11172417521 5.14338865452 2.08906220812 2.99441286435 2.01587600856 2.17880830745 -1.59770413783 1.94932239296 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8.83929916313 13.3835183495 8.78572677267 11.6453852715 8.69709853426 7.9535709979 9.29787863226 9.30690372887 8.0589218965 8.70493330922 10.5073691102 8.30478030863 12.3809818194 7.97077690576 9.79679662955 13.2547723983 10.639500161 9.09472455915 10.6522218641 14.8748869767 13.7515640122 7.82907632143 12.2814419408 10.5660914214 10.6904045282 14.449763402 7.14641505605 7.89097081444 7.08131820221 10.8330554784 5.41023092699 10.1558205558 9.07829801421 8.58325443421 11.1517512665 9.19527635526 6.66690915558 8.46208965901 7.62738974087 10.0574140554 10.245329046 8.52598026859 11.5061267753 5.38443291102 10.5519059704 8.4366143597 7.74718681847 10.1162822164 12.147985194 12.2570570598 6.36530323068 5.46735671995 8.70497885889 10.2401716107 9.05212092746 7.65748679322 9.40819854251 8.82478202856 13.3288667957 10.4607063022 8.03242140803 10.1313928308 10.5140999796 12.497383358 11.1964811646 5.8440220459 9.85220709149 10.7053549054 7.37322744116 10.0916871427 9.68655484795 8.75078674429 8.68075907116 10.0943818982 9.84259026664 9.61829674876 9.495559834 10.9643035269 7.56443438431 8.01986213902 9.12888645046 9.7883380811 11.881303172 11.4541501938 13.1505106945 9.73593440273 11.4762740768 11.489786996 3.32934010142 12.5756818595 6.49206974641 10.4037683338 14.3436934452 14.6497467536 15.590026453 8.75116521617 8.74636778222 8.52262659275 14.1458043596 8.73091348729 10.1507843828 9.24788254834 8.80177817145 12.3428676658 8.95969926172 9.32404134625 6.91897724226 11.2313909288 12.0887814966 8.29012075287 7.94820642447 16.0807762985 7.91126505228 13.636599612 11.583708125 9.29179040602 11.9573427441 8.65997498502 8.8170697274 13.7126709538 10.3692507396 9.35048728365 8.19135652597 8.58216610133 8.26001540331 10.3491122515 8.34527144043 9.49639600245 8.356540639 11.4176629242 12.8840278734 10.8342207365 9.56560798605 7.52888077095 9.62605485496 7.76551294502 9.72956337764 7.41823443489 11.1503664176 12.9015543456 13.8326311225 8.69538339078 10.3009067785 10.4821188162 11.1405377477 12.7949552436 10.1560630388 8.85777809106 8.04026161331 8.14345285276 7.90363524075 9.03825970446 10.4320912211 9.08304480712 6.71809165564 12.1967376127 8.83386848532 10.7821652988 10.2051655431 11.0944577888 9.32871765372 11.4059022951 11.4773821314 10.1700465248 12.2365730783 9.38963758344 8.26625206268 8.09857204544 7.23119333898 9.61801072172 7.29582817589 11.3834810873 7.99005155778 10.2359491589 9.87417900864 8.6651768565 9.95533207404 9.55128991272
Expected Output:
| **italy -> italian** :: | spain -> spanish |
| **india -> delhi** :: | japan -> tokyo |
| **man -> woman ** :: | boy -> girl |
| **small -> smaller ** :: | large -> larger |
Once you get the correct expected output, please feel free to modify the input cells above to test your own analogies. Try to find some other analogy pairs that do work, but also find some where the algorithm doesn't give the right answer: For example, you can try small->smaller as big->?.
You've come to the end of this assignment. Here are the main points you should remember:
Even though you have finished the graded portions, we recommend you take a look too at the rest of this notebook.
Congratulations on finishing the graded portions of this notebook!
In the following exercise, you will examine gender biases that can be reflected in a word embedding, and explore algorithms for reducing the bias. In addition to learning about the topic of debiasing, this exercise will also help hone your intuition about what word vectors are doing. This section involves a bit of linear algebra, though you can probably complete it even without being expert in linear algebra, and we encourage you to give it a shot. This portion of the notebook is optional and is not graded.
Lets first see how the GloVe word embeddings relate to gender. You will first compute a vector $g = e_{woman}-e_{man}$, where $e_{woman}$ represents the word vector corresponding to the word woman, and $e_{man}$ corresponds to the word vector corresponding to the word man. The resulting vector $g$ roughly encodes the concept of "gender". (You might get a more accurate representation if you compute $g_1 = e_{mother}-e_{father}$, $g_2 = e_{girl}-e_{boy}$, etc. and average over them. But just using $e_{woman}-e_{man}$ will give good enough results for now.)
g = word_to_vec_map['woman'] - word_to_vec_map['man']
print(g)
Now, you will consider the cosine similarity of different words with $g$. Consider what a positive value of similarity means vs a negative cosine similarity.
print ('List of names and their similarities with constructed vector:')
# girls and boys name
name_list = ['john', 'marie', 'sophie', 'ronaldo', 'priya', 'rahul', 'danielle', 'reza', 'katy', 'yasmin']
for w in name_list:
print (w, cosine_similarity(word_to_vec_map[w], g))
As you can see, female first names tend to have a positive cosine similarity with our constructed vector $g$, while male first names tend to have a negative cosine similarity. This is not suprising, and the result seems acceptable.
But let's try with some other words.
print('Other words and their similarities:')
word_list = ['lipstick', 'guns', 'science', 'arts', 'literature', 'warrior','doctor', 'tree', 'receptionist',
'technology', 'fashion', 'teacher', 'engineer', 'pilot', 'computer', 'singer']
for w in word_list:
print (w, cosine_similarity(word_to_vec_map[w], g))
Do you notice anything surprising? It is astonishing how these results reflect certain unhealthy gender stereotypes. For example, "computer" is closer to "man" while "literature" is closer to "woman". Ouch!
We'll see below how to reduce the bias of these vectors, using an algorithm due to Boliukbasi et al., 2016. Note that some word pairs such as "actor"/"actress" or "grandmother"/"grandfather" should remain gender specific, while other words such as "receptionist" or "technology" should be neutralized, i.e. not be gender-related. You will have to treat these two type of words differently when debiasing.
The figure below should help you visualize what neutralizing does. If you're using a 50-dimensional word embedding, the 50 dimensional space can be split into two parts: The bias-direction $g$, and the remaining 49 dimensions, which we'll call $g_{\perp}$. In linear algebra, we say that the 49 dimensional $g_{\perp}$ is perpendicular (or "othogonal") to $g$, meaning it is at 90 degrees to $g$. The neutralization step takes a vector such as $e_{receptionist}$ and zeros out the component in the direction of $g$, giving us $e_{receptionist}^{debiased}$.
Even though $g_{\perp}$ is 49 dimensional, given the limitations of what we can draw on a screen, we illustrate it using a 1 dimensional axis below.

Exercise: Implement neutralize() to remove the bias of words such as "receptionist" or "scientist". Given an input embedding $e$, you can use the following formulas to compute $e^{debiased}$:
If you are an expert in linear algebra, you may recognize $e^{bias\_component}$ as the projection of $e$ onto the direction $g$. If you're not an expert in linear algebra, don't worry about this.
def neutralize(word, g, word_to_vec_map):
"""
Removes the bias of "word" by projecting it on the space orthogonal to the bias axis.
This function ensures that gender neutral words are zero in the gender subspace.
Arguments:
word -- string indicating the word to debias
g -- numpy-array of shape (50,), corresponding to the bias axis (such as gender)
word_to_vec_map -- dictionary mapping words to their corresponding vectors.
Returns:
e_debiased -- neutralized word vector representation of the input "word"
"""
### START CODE HERE ###
# Select word vector representation of "word". Use word_to_vec_map. (≈ 1 line)
e = None
# Compute e_biascomponent using the formula give above. (≈ 1 line)
e_biascomponent = None
# Neutralize e by substracting e_biascomponent from it
# e_debiased should be equal to its orthogonal projection. (≈ 1 line)
e_debiased = None
### END CODE HERE ###
return e_debiased
e = "receptionist"
print("cosine similarity between " + e + " and g, before neutralizing: ", cosine_similarity(word_to_vec_map["receptionist"], g))
e_debiased = neutralize("receptionist", g, word_to_vec_map)
print("cosine similarity between " + e + " and g, after neutralizing: ", cosine_similarity(e_debiased, g))
Expected Output: The second result is essentially 0, up to numerical roundof (on the order of $10^{-17}$).
| **cosine similarity between receptionist and g, before neutralizing:** : | 0.330779417506 |
| **cosine similarity between receptionist and g, after neutralizing:** : | -3.26732746085e-17 |
Next, lets see how debiasing can also be applied to word pairs such as "actress" and "actor." Equalization is applied to pairs of words that you might want to have differ only through the gender property. As a concrete example, suppose that "actress" is closer to "babysit" than "actor." By applying neutralizing to "babysit" we can reduce the gender-stereotype associated with babysitting. But this still does not guarantee that "actor" and "actress" are equidistant from "babysit." The equalization algorithm takes care of this.
The key idea behind equalization is to make sure that a particular pair of words are equi-distant from the 49-dimensional $g_\perp$. The equalization step also ensures that the two equalized steps are now the same distance from $e_{receptionist}^{debiased}$, or from any other work that has been neutralized. In pictures, this is how equalization works:

The derivation of the linear algebra to do this is a bit more complex. (See Bolukbasi et al., 2016 for details.) But the key equations are:
$$ \mu = \frac{e_{w1} + e_{w2}}{2}\tag{4}$$ $$ \mu_{B} = \frac {\mu \cdot \text{bias_axis}}{||\text{bias_axis}||_2^2} *\text{bias_axis} \tag{5}$$ $$\mu_{\perp} = \mu - \mu_{B} \tag{6}$$$$ e_{w1B} = \frac {e_{w1} \cdot \text{bias_axis}}{||\text{bias_axis}||_2^2} *\text{bias_axis} \tag{7}$$$$ e_{w2B} = \frac {e_{w2} \cdot \text{bias_axis}}{||\text{bias_axis}||_2^2} *\text{bias_axis} \tag{8}$$
$$e_{w1B}^{corrected} = \sqrt{ |{1 - ||\mu_{\perp} ||^2_2} |} * \frac{e_{\text{w1B}} - \mu_B} {|(e_{w1} - \mu_{\perp}) - \mu_B)|} \tag{9}$$$$e_{w2B}^{corrected} = \sqrt{ |{1 - ||\mu_{\perp} ||^2_2} |} * \frac{e_{\text{w2B}} - \mu_B} {|(e_{w2} - \mu_{\perp}) - \mu_B)|} \tag{10}$$$$e_1 = e_{w1B}^{corrected} + \mu_{\perp} \tag{11}$$$$e_2 = e_{w2B}^{corrected} + \mu_{\perp} \tag{12}$$Exercise: Implement the function below. Use the equations above to get the final equalized version of the pair of words. Good luck!
def equalize(pair, bias_axis, word_to_vec_map):
"""
Debias gender specific words by following the equalize method described in the figure above.
Arguments:
pair -- pair of strings of gender specific words to debias, e.g. ("actress", "actor")
bias_axis -- numpy-array of shape (50,), vector corresponding to the bias axis, e.g. gender
word_to_vec_map -- dictionary mapping words to their corresponding vectors
Returns
e_1 -- word vector corresponding to the first word
e_2 -- word vector corresponding to the second word
"""
### START CODE HERE ###
# Step 1: Select word vector representation of "word". Use word_to_vec_map. (≈ 2 lines)
w1, w2 = None
e_w1, e_w2 = None
# Step 2: Compute the mean of e_w1 and e_w2 (≈ 1 line)
mu = None
# Step 3: Compute the projections of mu over the bias axis and the orthogonal axis (≈ 2 lines)
mu_B = None
mu_orth = None
# Step 4: Use equations (7) and (8) to compute e_w1B and e_w2B (≈2 lines)
e_w1B = None
e_w2B = None
# Step 5: Adjust the Bias part of e_w1B and e_w2B using the formulas (9) and (10) given above (≈2 lines)
corrected_e_w1B = None
corrected_e_w2B = None
# Step 6: Debias by equalizing e1 and e2 to the sum of their corrected projections (≈2 lines)
e1 = None
e2 = None
### END CODE HERE ###
return e1, e2
print("cosine similarities before equalizing:")
print("cosine_similarity(word_to_vec_map[\"man\"], gender) = ", cosine_similarity(word_to_vec_map["man"], g))
print("cosine_similarity(word_to_vec_map[\"woman\"], gender) = ", cosine_similarity(word_to_vec_map["woman"], g))
print()
e1, e2 = equalize(("man", "woman"), g, word_to_vec_map)
print("cosine similarities after equalizing:")
print("cosine_similarity(e1, gender) = ", cosine_similarity(e1, g))
print("cosine_similarity(e2, gender) = ", cosine_similarity(e2, g))
Expected Output:
cosine similarities before equalizing:
| **cosine_similarity(word_to_vec_map["man"], gender)** = | -0.117110957653 |
| **cosine_similarity(word_to_vec_map["woman"], gender)** = | 0.356666188463 |
cosine similarities after equalizing:
| **cosine_similarity(u1, gender)** = | -0.700436428931 |
| **cosine_similarity(u2, gender)** = | 0.700436428931 |
Please feel free to play with the input words in the cell above, to apply equalization to other pairs of words.
These debiasing algorithms are very helpful for reducing bias, but are not perfect and do not eliminate all traces of bias. For example, one weakness of this implementation was that the bias direction $g$ was defined using only the pair of words woman and man. As discussed earlier, if $g$ were defined by computing $g_1 = e_{woman} - e_{man}$; $g_2 = e_{mother} - e_{father}$; $g_3 = e_{girl} - e_{boy}$; and so on and averaging over them, you would obtain a better estimate of the "gender" dimension in the 50 dimensional word embedding space. Feel free to play with such variants as well.
You have come to the end of this notebook, and have seen a lot of the ways that word vectors can be used as well as modified.
Congratulations on finishing this notebook!
References: